Theorem f1eq3 2778

Description: Equality theorem for one-to-one functions.

Assertion

Ref	Expression
f1eq3	|- (A = B -> (F:C-1-1->A <-> F:C-1-1->B))

Proof of Theorem f1eq3

Step	Hyp	Ref	Expression
1		feq3 2750	. . 3 |- (A = B -> (F:C-->A <-> F:C-->B))
2	1	anbi1d 469	. 2 |- (A = B -> ((F:C-->A /\ Fun `'F) <-> (F:C-->B /\ Fun `'F)))
3		df-f1 2435	. 2 |- (F:C-1-1->A <-> (F:C-->A /\ Fun `'F))
4		df-f1 2435	. 2 |- (F:C-1-1->B <-> (F:C-->B /\ Fun `'F))
5	2, 3, 4	3bitr4g 428	1 |- (A = B -> (F:C-1-1->A <-> F:C-1-1->B))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091  `'ccnv 2409  Fun wfun 2416  -->wf 2418  -1-1->wf1 2419

This theorem is referenced by:  f1oeq3 2797  brdomg 3281

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-in 1491  df-ss 1492  df-f 2434  df-f1 2435

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